Apparatus for processing angular data

ABSTRACT

Resolver apparatus is disclosed employing a system which receives resolver inputs Rsin theta and Rcos theta and is adjusted according to coarse and fine processor angles Alpha c and Alpha f to achieve a measurement of, or facilitate operations on, the resolver input angle theta . The apparatus employs two computing channels, one for each resolver input, a function generator circuit for developing tangent (or cotangent) functions of Alpha c, a linear interpolator in one of the channels, and circuits interconnecting the channels so that the signals ultimately developed in the channels bear a relationship which permits one of them to be linearly interpolated as a function of Alpha f while maintaining an interpolating angle error characteristic which is small and independent of Alpha c. A simple passive compensating technique may be employed to take advantage of this characteristic to produce a highly accurate response. Three of the embodiments employ tapped and switched dividers to generate the tangent functions while a fourth exploits those values of Alpha c which reduce the tangent functions to zero of unity, thereby permitting the use of simple combining circuits for function generation.

United States Patent Bifulco, Jr. 1 Feb. 29, 1972 [54] APPARATUS FOR PROCESSING ANGULAR DATA [57] ABSTRACT [72] Inventor: Harry P. Bifulco, .lr., Huntington, N.Y. Resolver apparatus is disclosed employing a system which receives resolver inputs Rsin0 and Rcosl) and is adjusted ac- [73] Assignee. gozth Atlantic Industries Inc., Plainvrew, cording to coarse and fine processor angles at and a, to

achieve a measurement of, or facilitate operations on, the

[22] Filed: Sept. 29, 1969 resolver input angle 0. The apparatus employs two computing channels, one for each resolver input, a function generator cir- [211 Appl' 861379 cuit for developing tangent (or cotangent) functions of a a linear interpolator in one of the channels, and circuits inter- [52] U.S.Cl ..235/l86, 318/637, 318/654, connecting the channels so that the signals ultimately 313/661 developed in the channels bear a relationship which permits [Ill-CL G0 8 /22 one of them to be linearly interpolated as a function of a,

[ Field of Search "340/347 SY; 318/637, 6 while maintaining an interpolating angle error characteristic 323/435; 235/ 186, 1 which is small and independent of a A simple passive compensating technique may be employed to take advantage of [56] Reign this characteristic to produce a highly accurate response.

UNITED STATES PATENTS Three of the embodiments employ tapped and switched dividers to generate the tangent functions while a fourth ex- 3,250,905 1966 hl' ol' cl 1 347 SY ploits those values of a, which reduce the tangent functions to Primary ExaminerEugene G. Botz Assistant Exami zero of unity, thereby permitting the use of simple combining circuits for function generation.

nerR. Stephen Dildine, Jr. Attorney-Morgan, Finnegan, Durham & Pine 64 Claims, 4 Drawing Figures 50 2| x, 2 8 =RS|n9 L Cos =RS 0 es m e =RCos9TonHd 1p 1 I 2 7 l 20 XPZ I? ee=e (f' 5299995.- l 3 cos- 0 1:

time P N: l RSma I V i, w XP' I ol'lnull e mcosa Sb t S|n(D-A() I SmAx l c e =RCos9 Ten t; a zk A,

4 a t R u o t P x, +0 ,Tc 6

PAIENTEDFB29 m2 SHEET 1 [IF 4 hv mixq mum m INVENTOR. y R Bifu|co,Jr.

ATTORNEYS APPARATUS FOR PROCESSING ANGULAR DATA This invention relates to bridge apparatus which receives input angular data, for example, resolver input data in the form RsinH and RcosO, and which processes same, e.g., by computing the value of in digital form.

Since a resolver bridge is in essence a control transformer, the term resolver bridge as used herein contemplates input angle measuring bridges, deviation-type bridges, programmed bridges including those used for coordinate rotation and differential resolver functions, and systems providing other functions typical of control transformer applications such as those relating to phase shifting and the-like.

In a typical direct angle reading arrangement, the system is adjusted by a processor anglea and is instrumented such that when a predetermined condition exists, the processor angle is equal to the input angle and is therefore a measurement of same.

The inputs to angle-measuring resolver bridge apparatus consist of two input functions Rsin0 and RcosO each having two unknowns or variables, amplitude R and angle 6. It is therefore necessary to process two computing functions of the input variables and the processor angle a, thereby to obtain the correct value of the input angle 0 or to perform some operation thereon.

For example, when the function Rcosfltana equals the function RsinO, then the processor angle a indicates and is equal to the input angle 0.

ln mechanizing solutions to obtain a direct indication of input angle 0, it is necessary to generate trigonometric functions of the angle a. The function tan a, above, is one example, another is found in the system which utilizes the functions:

F,=Rcos0sina F Rsinllcos a When F equals F 0: equals Here the processor angle functions sin a and'cosa must be generated.

in both of the above examples, the functions of the processor angle a are trigonometric. Such functions can give rise to limitations in systems which are required to compute the input angle 0 to a high degree of accuracy for a multiplicity of values, e.g., 100,000 increments. In such systems the processor angle must be adjustable to the same degree of resolution. While this order of resolution can readily be achieved in generating linear functions, practical limitations involving cost, size, weight and complexity are encountered in achieving comparable resolution in the generation of trigonometric functions.

One approach to this problem relies on the above relationship 0=a, when Rsin0cosa=Rcos0sina, and utilizes trigonometric function generators of a coarse processor angle, together with linear interpolating techniques, to refine the coarse angle functions with smaller angular increments to achieve the com and sina terms. For example, one can generate sine and cosine functions of coarse adjustment angles which occur in, say, lO-degree increments, e.g., 0, 20 and then interpolate each of the two functions over the coarse angle increments with successively smaller fine angle increments utilizing precision dividers. However, these dividers normally interpolate linearly rather than trigonometrically, thereby introducing an error. For example, the sinusoidal voltage e=sinu does not vary linearly with a over any interval and moreover the departure from linearity depends upon the location of the particular interpolating interval.

Compensation for this error is thus complicated becausethe amount of compensation depends on the particular interpolating interval.

An effective solution to this problem is disclosed in the copending application of John Heaviside, filed Mar. 20, 1964 Ser. No. 353,558 now Pat. No. 3,493,735 and entitled, Computer Circuits For Processing Trigonometric Data. Disclosed therein is an interpolating technique which exploits the readily obtained high precision and resolution of linear interpolating dividers while at the same time limiting interpolation error. The technique also facilitates simple compensation for those errors that do exist.

The technique disclosed therein involves the establishment of a coarse angle value ac" (*The Heaviside application uses a somewhat different notation scheme.) which in general is different from the input angle 0, the difference being Au where a has the value immediately below 0. The coarse angle modifies the input functions of R and 0 to facilitate the derivation of two computing signals. Both signals in the illustrative embodiment of Heaviside are functions of R and their ratio is independent of R and a function of the variable, Aa, above.

The two computing signals, designated e, and e, in the Heaviside application, are derived to have characteristics such that one of them, e,, can be interpolated using cascaded linear dividers adjusted by a set of fine processor angles of successively smaller increments totaling 01,. This part of the processing operation may be regarded as multiplying the signal e,- by a linear function of a, to obtain a new signal containing the product a,.e,, which signal is so related to the other original signal e that when they are equal, variable R vanishes and a' closely approaches a linear relationship to Aa.

Accordingly, the net processor angle adjustment 0:, which is the sum [a +a,], approaches true proportionality with the input angle 0 and, with proper calibration, thereby indicates the value of 6.

The above system provides several important advantages, one being the need to interpolate only one function and another being the relative ease with which the interpolated function may be compensated because the interpolation angular error is independent of the coarse angle setting.

The present invention represents further improvements in the foregoing technique and has for an object the attainment of material circuit simplifications, realizable without loss of the above-mentioned constant error characteristic.

It is another object of this invention to simplify the switching arrangements in resolver bridge systems while preserving high accuracy and resolution.

A further object of the invention is to configure the bridge system to exploit the capabilities of operational amplifiers.

A still further object of the inventionis to provide a resolver bridge system adaptable to large scale integration techniques.

Other objects of the invention are to provide a bridge (1) which is operable over a wide range of frequencies including DC, (2) which eliminates the shortcomings associated with inductive elements, and (3) which by virtue of their absence is particularly amenable to automatic operation.

Other objects and advantages of the invention will be apparent in the description which follows.

Serving to illustrate exemplary embodiments of the invention are the drawings of which:

FIG. 1 is a schematic diagram of a first embodiment of the invention;

FIG. 2 is a schematic diagram of a second embodiment utilizing operational amplifiers for performing certain functions;

FIG. 3 is a schematic diagram illustrating a third embodiment of the invention.

FIG. 4 is a schematic diagram of a system employing the general technique of FIG. 3 in an automatic bridge which is also capable of processing synchro inputs, and which is operable without the need for tapped dividers.

The system of FIG. 1 receives resolver inputs in the form of a voltage e,=Rsin0, applied to input terminals l, and I of channel 1 and a second voltage e =Rcos0 which appears at input terminals I and of channel 2. These inputs are operated on with functions of a coarse processor angle a and fine processor angle a,, derived from respective switching control units T and T such that the sum of the values a, and a; necessary to establish a null condition, is equal to the input resolver angle 0.

The resolver inputs e, and e,. are connected to respective arms of two ganged switches S and S the sine input e, appearing at S and the cosine input e at switch 8,, The switches have two positions and are controlled by the coarse processor angle 01 to be in the position shown in the range 315 to 45 and to 225, and to be switched to the alternate position in the other intervals.

The data flow described below will represent the case where the processor angle 01 is in the first mentioned ranges 315 to 45 or 135 to 225, it being understood that the resolver in puts are effectively interchanged and other switching effected in the interval 45 to 135 and 225 to 315 as explained more fully hereinafter.

Under the illustrated conditions, the sine resolver input e, is connected to the primary XP of a transformer X2 while the cosine resolver input e is connected to an inductive divider Dl having a group of taps T T,,,, T arranged in quarter octant (11.25") increments. Switchable contacts S1, and S1,, are controlled by the coarse processor angle (1,. derived from T thereby selectively contacting two adjacent taps T T then T T etc., to thereby respectively simulate tan 01 and tan (a +D) where D represents the coarse angle increment of ll.25. The same simulation occurs in the range 135 to 225 while in the other intervals, the divider D1 generates the function cotangent a with the switching proceeding in the reverse (downward) order.

Since D1 has been energized by RcosO in the assumed case, a voltage e appears at arm S1, of the divider which represents the function RcosO tana while at S], a voltage e representing thefunction Rcos tan(a +D) is generated.

The voltage e, appearing at arm S1,, is applied to one terminal 20, of the primary XP of the transformer X2 while, as previously noted, the other end 21 of the primary is connected to the resolver sine input e, via switch S These two signals e, and e are thus subtracted with the difference appearing across the secondary XS of the transformer X2.

The secondary X5 is also tapped at l l.25 increments with the taps T T m, being selectively contacted by the switch arm S2,, in accordance with the value of the selected coarse angle. The tap distribution is adjusted according to the function cos (a D) cos a,

As a consequence, over the ranges 3l5 to 45 and 135 to 225", there appears in channel 1 at the arm of S2,, relative to the end terminal 24, a voltage e;,, where:

R sin (D- Aa) e3 cos or,

The factor Aa represents the coarse error, i.e., 0-11 This voltage constitutes a part of the input signal to a linear interpolator lP which may take any one of a number of suitable forms, e.g., known cascaded resistive dividers, or switched operational amplifier arrangements as described hereinafter.

Referring back to the divider D1, it may be seen that the switch arm S1,, which develops the signal Rcos0tana is connected to one end of the primary XP, of a transformer XI. The other terminal 11 of the primary is energized by the resolver input, Rsin0, so that the net primary voltage represents the difference between these two quantities. This difference appears across the secondary XS of X1 and over the ranges mentioned above is the function:

R sin Au COS u Since one terminal of the interpolator IP is grounded as is one terminal of secondary XS the net voltage applied to the interpolator represents the sum of (1) the voltage 2 appearing at S2,, across X8 and (2) the voltage e. across the secondary XS of X1. This sum is accordingly representative of the function: g

R[sin (D-Aa) sin Aa] cos 01,

The interpolating circuit IP is adjusted by the setting of the fine angle, a,, and includes means, schematically indicated by c, for compensating the linear interpolator to account for the interpolating error.

where F is defined as the null function or Ol 2: ICAOI.

i s es tantsqsffisiqnt- Since under these conditions the fine angle a, is closely proportional to the difi'erence between the input angle 0 and the coarse processor angle a i.e. to Act, then the sum of the coarse and fine processor angle values equals the input angle 0 within the error characteristic described below. In other words,

at null arr-(Fa and By reading the null values of the coarse angle as generated at switch control T and the fine angle generated by switch control T the angle 0 may be indicated.

An analysis of the foregoing arrangement will show that without compensation C, the system angular error due to the use of a linear interpolator lP varies in S-shape'd manner over the interval D and is independent of the coarse angle a as described in the aforesaid Heaviside application. The error 40 is zero at a,=0, D/2 and D and in the case of D=l l.25, has a maximum value of about 25" at approximately the D/4 and 3D/4 points. Correction may be made in interpolator [P at the nonzero interpolation points occurring over the 1 125 interval, greatly reducing system error. This correction can be introduced for example by varying the linearity of the interpolator, as by changing divider tap position; or, where operational amplifiers are used, the values of certain of the input resistances may be modified over the values they would have in a purely linear case.

Considering now the intervals 45 to 135 and 225 to 315, analysis will show that the null function F, is still applicable, wh le t e gtherfuns ion a t ans ted, t"? ti The relationships are as follows:

The means for generating the cofunctions is readily apparent by observing the switching performed by Sa, Sb and by noting that the tap arrangement and switching of divider Dl relies on the equality between tanac and cotan(90a,). Thus It may be noted that system sensitivity as the null is ap-.

proached varies as a function of l/cosor or l/sinor depending on the selected intervals. 7 I

To illustrate that the described technique is applicable to both manual and automatic bridges, a feedback system FB coupled between the null detector and the coarse and fine controls T and T, is shown. The signal e, is available for other applications and modes of operation including those mentioned in the introduction.

FIGURE 2 To facilitate switching to ground in the function generation circuits, the embodiment of FIG. 2 generates tangent functions of the coarse trial angle in each channel; it also uses operational amplifiers for combining signals, for scaling and for linear interpolation, and employs single tap selection in the tangent function generators.

The description which follows applies to the case where 0 is in the first, fourth, fifth or eight octants. The sine resolver input 2,, appearing between terminals I and is applied to the primary of a transformer X4. This transformer includes two tapped secondaries X84, and XS4,, functionally in channels 1 and 2, respectively. The cosine resolver input e appearing between input terminals 1 and 1 is applied to the primary of transformer X3. This transformer also includes two tapped secondaries XS3, and XS3,, functionally operative in channels 2 and 1, respectively.

The secondaries XS3, and X84, are tapped and selectively switched to ground to generate tangent or, functions at 11.25 intervals. As a consequence, a'voltage e =Rcos0 tana, appears at the output of XS3, and a voltage e,,=Rsin0 tan a, is developed across XS4,. The voltage e is combined with the voltage e,=Rsin0 induced in the secondary X84, of X4 yielding the voltage R sin Ad 12 The secondary XS4, includes reversing switches S4,, S4,, S4 and S4, which in effect interchange the input and output terminals of the secondary as (1, goes from one octant to the next to provide the required sign changes. The switches are also operative in developing the cofunctions generated in the second, third, sixth and seventh octants as described hereafter.

The end terminals of XS4, may be alternately connected to the input of operational amplifier 0P depending on the selected coarse angle octant, via the switches S4, and S4, which are always in opposite states.

The input to XS4, is derived from the lower end terminal of XS3, which is connected to either the upper or lower end terminal of X84, depending upon the position of the switches S4 and S4,. in the position shown, the lower terminal of XS3, is connected to the lower terminal of XS4, while the upper terminal of X84, is connected to operational amplifier OP in channel 1 a comparable arrangement is employed. Voltage e across XS4, is combined with the input voltage e,= cos0 by coupling X84, to secondary XS3,, the latter having induced therein the input voltage e,. A reversing arrangement similar to that employed with XS4, characterizes the switching of XS3,. Its end terminals are alternatively connected via respective switches S3, and S3, to the bottom terminal of secondary X84, and via respective switches S3 and S3, to the common terminal (ground). in the condition shown the lower terminal of XS3, is connected to the common terminal while its upper terminal is connected to the lower terminal of XS4,.

The combined output of XS3, and X54, constituting e e,+e is coupled via the selected tap of X54, to the input of operational amplifier 0P2.

By virtue of the foregoing arrangements, voltages e and e are generated in the two channels such that:

R sin A01 Operational amplifier 0P1 which receives voltage e provides the required sign change and illustratively includes a scaling network comprising resistors R20 and R21 connected in series and having their junction returned as an input to OH. The remote end of R20 is connected to the output of GP] while the remote end of R21 is connected to ground. A similar network comprising resistors R22 and R23 is employed for scaling the signal g applied to operational amplifier 0P2 and inverted thereby. As in the case of the other embodiments, system scaling does not change the computational technique; its affects on coefficients of the processed signals are accordingly not shown in the equations used herein. Sign changes are not shown for the same reason. Thus, e at the output of 0P1 is described as being the same as e The effects of sign change and system scaling on e are similarly not, shown.

The output e from 0P1 is applied via summing resistor R24 to a summing amplifier 0P4 where it is summed with another input representing the interpolated output of 0P2.

The output of 0P2 is applied to an operational amplifier -.OP3 having a feedback resistance R26 and a series of summing resistors R27 which can be switched into or out of the input circuit via associated switches S27 in various combinations controlled by the fine angle switching means T,-. As a result, the output of 0P3 represents the interpolated output of 0P2 and can be represented by:

= R cos Au tan Au 1 tan D W Since small angles are involved, tanAa closely approximates 01;, then a,-'==Aa=0a,. Accordingly, the sum of the null values of the coarse and fine processor angles a and a, approximates the input angle 0 to be measured. Compensation can be effected conveniently in the linear interpolator following the procedure described in connection with FIG. 1.

For coarse angle interpolating interval D of l 1.25, the interpolating error in the F 16. 2 system varies from zero at D=O to a maximum of about 200 minutes at approximately BIZ and thence back to zero at D=11.25. This C-shaped error function is the same as the S-shaped error function over half the S curve interpolating interval characterizing FIG. 1 and also described in the Heaviside application, it being understood that the interval D of the latter is twice that of the former.

As noted previously, the foregoing description applies when 0 lies in the first, fourth, fifth and eighth octants (3l545;

-225). In the other octants, the cofunctions listed below i are generated by appropriate coarse an gle switchmg circuits R81 and by operation of the sign-changing switching previously described. Switch circuit RS1 includes a channelreversing switch for exchanging the inputs to 0P1 and 0P2 system of FIG. 2 to obtain the embodiment shown in FIG. 3.

e a; tan D 7 l While the errors in the circuit of FIG. 2 are low for the amount of hardware involved, still greater accuracy may be obtained along with significant economies by modifying the l 5 FIGURE 3 The system of FIG. 3 is identical with that of FIG. 2 in the operations performed on the resolver inputs e, and e, by the coarse trial angle functions tan a (and cotan a to derive the functions of e and e The mechinization differs thereafter: First, as can be seen in FIG. 3, the scaling networks associated with CPI and 0P2 have been eliminated to provide unity gain and the outputs of these two amplifiers are combined before interpolation in an operational amplifier 0P5. This amplifier receives the output of 0P1 via summing resistor R and the output of 0P2 via summing resistor aR. The feedback resistor of CPS is set at a value equal to aR and a is set equal to 1- cosD)/sinD with the result that the net output from 0P5 represents the function:

R (sin (DAa)+sinA a) sinD e 0 813 "i' Z1812 COS 110 This function is then interpolated by applying voltage e to COS Ol e [sin (D-Aa) sin Au] dr This is the same function that is represented by e of FIG. 1 and yields the. same S-shaped interpolating error characteristics. r

As in the case of FIG. 2, the output, e of 0P1 is summed with e in summer 0P4 via the resistance R24. Stage 0P4 includes a feedback resistor R30, and a resistance R31 connected from the summing junction side of R25 to ground for sealing.

When the coarse and fine trial angles are adjusted to a null condition at the output of 0P4, then e =e (excluding scaling effects) and i R sin Aa R(sin D- Aa) -lsin Au) a COS 0&- cos s f Accordin l Sin (D A...

in A w-..

To generate the required cofunctions in the second, third, sixth and seventh octants, the system of FIG. 3 employs the switching techniques of FIG. 2.

In the embodiments heretofore described it has been necessary to employ tapped dividers with tap distribution and tap :selection controlled to provide the tana (or cotana functions.

The system equations controlling the embodiments of FIGS. 2 and 3 in an illustrative interval are:

I=Cos!) Sin!) Tana: and 'li. CPQI v where the function I is generated in the interpolation channel and S in the other channel (not to be confused with the S- shaped error function).

By interpolating I as a function of a, and nulling the resultant with the S signal, the angle 0 was computed in the 'embodiment of FIG. 2 while in the case of FIG. 3 the I signal was modified by the S signal before interpolation to improve the error characteristic.

The embodiment of FIG. 4 is based on the recognition that: (I) the tangent factor in the above listed I and S functions reduces to the values 0 or I where a, is restricted to 0 and multiples of 45; (2) that the relatively large error associated with a 45 interpolation interval can be reduced by interpolating t22.5 in an up/down fashion about the coarse angle setting; and (3) still further reduction in error can be achieved by utilizing the cross coupling techniques of FIG. 3.

As a consequence of item 1) above, no dividers or taps are required, the generator of I & S being reduced to simple combinations of the input functions Cos 0 and Sin 0.

Thus, for the values a =0 and 45, the l & S functions become:

I=Cos 0 I=Cos O-I-Sin 0 S=Sin 0 S=Sin 0-Cos 0*(*'I'he factor R has been omitted from the following discussions for the sake of simplification.)

This approach thus permits the generation of the I & S functions by simple summing circuits without the need for trigonometric dividers or other trigonometric computing elements while at the same time providing the low error features.

FIGURE 4 The system of FIG. 4 combines a bridge with a converter and automatic feedback system. The converter CV40 is designed to adapt the resolver bridge BR40 to the processing of synchro data and to this end converts the data in synchro form to equivalent data represented'in resolver form.

The feedback system includes a counter CN40 and decoding circuits L40, L41 for controlling the adjustments of the coarse and fine processor angles in response to a signal derived from the null circuit NC40. By virtue of this arrangement an automatic conversion of synchro data to equivalent digital data is provided.

The general data flow is initiated at the input terminals Y Y and Y;, which receive the synchro voltages. These synchro voltages are converted in the synchro-to-resolver converter CV40 into equivalent resolver voltages which are applied in turn to the bridge circuit BR40.

The bridge employs a function generator FG40 and linear interpolator IP40 adjusted respectively by a coarse angle input a and a fine angle input 01,, and drives a null detector NC40.

The null detector operates a BCD counter CN40 which controls in turn the values of the coarse and fine processor angles via the decoders L40 and L41 respectively.

When a null condition has been established, the state of the 0 counter CN40 provides a digital representation of the input This function is the same as the one obtaining with respect to FIG. I. The interpolation error is of the S-shaped type described in connection with FIG. I and the previously mentioned Heaviside application; it has a maximum value of about 25" for D=l l.25, which is lath that of the FIG. 2 circuit. By modifying the total input resistance in the interpolator 0P3, the error is readily compensated.

synchro angle which may be read out as indicated. To digitize resolver inputs, converter CV40 is bypassed.

The synchro-to-resolver converter CV40 employs two channels, one comprising the cascaded operational amplifiers 0P40 and OP4I, and the other, the similarly cascaded amplifiers 0P50 and 0P5l.

The input operational amplifier OP40 receives the voltages 9 l appearing at terminals Y and Y via the respective surnrnin g m computational techniques following that previously described resistors R40 and R41. The amplifier is provided with a feedat the end of the description of AS noted there, the back resistance R42. The output of OP40 is coupled via R43 coarse angle c dividers have been replaced y m ng to the input ofOP4l which also includes afeedback resistance 5 P!i ?9 R44.

The output of 0P4] is supplied to the bridge circuit and, One of the sets of coarse angle function generators comspecifically, to the switches 41 and 54 of the bridge function prises the switching circuits 41 through 44 which control the generator F640. The output of 0P40 also directly supplies th i insertion of the associated resistances R46 in circuit with the switching network of the bridge, being connected to the mming j nc i n of h p r i n l p fi r 0P42, the switch circuits 43 and 52. i values of the resistances being equal. The switches are actui lated by way of a control signal representing a which is derived from the decoder L40 which is controlled in turn by l the four most significant digits in the counter CN40.

The other channel of the converter, consisting of the am-i plifiers OPSO and 51 is supplied by the voltage appearing at Y, which through summing resistor R51 is coupled to OP50, and

the voltage appearing at Y which is coupled to OP50 via resistance R50. The summing junction side of R50 is returned tol The switching of the switching elements 41 through 44 and the input ground terminal Y via resistance R5 5. 7 g j the values of the associated resistances R46 are such as to Operational amplifier OP50 includes a feedback resistance generate the mgemac funcno'fs as was done R52 and has its output coupled via resistance R53 to the m case of the p'evlousfly descnbFd embodlmef'ts' These operational amplifier 0P5] having feedback resistance R54. ffmcnons. form Products the ismeo f icos'neo func The output of OP50 also directly supplies the switching cir- I denved m the y to yleld the func' wits 53 and 42 in the bridge function generator F640 trons. However, since or is restncted to values of zero and multiples of 45, the values of the tangent and cotangent factors reduce to zero or unity. The net functions and cofunctions thus become simply combinations of the resolver inputs :tsinO v a nd iCosO. V H The input summing and feedback resistances of the various operational amplifiers in the converter CV40 are assigned values required to provide the conversion of the input data Thus switches 41 and 43 are energlzed at the appropnate The output of 0P5l also feeds the coarse processor angle switching system of BR40, being connected to the switching circuits 51 and 44. 7

from synchro form to resolver form. Letting the values of R43, to supply oppositely phased i 0 functions 44 52 53 54 and 55 assume a resistance the remaining the switches 42 and 44 couple the oppositely phased cosine 0 resistances are assigned the following values to achieve this 'fupcuons when they energized The riislstanceswapd amconversion: I V plrfier 42 sum these signals to form various combination for A similar arrangement characterizes the switching elements R42=Rl2 I 51 through 54 and associated equal resistances R56 in the other channel of the bridge. Accordingly, the functions listed While a converter employing single-ended input stages has below appear at the outputs of the operational amplifier OP42 been shown, differential vut tes sntex 1 2 Psafihze- Qliflw it u r e Coarse S (OP42) I (OP52) angle signal signal I I 0 0-225 Sm 0 sin 0 COS cos 0 I 0s 0 cos 0 sin 5-45) cos 0-45 v 22.5-4-5 W (srn0 tilsll Sin cos0+sin0 figsw=sin B-cos 0 =cos 0+ sin 6 nmslnifim ,t,, s u ,t sin(090) cos(0-90) 7.5- 6 srn 00 cos 0 sin 90 Sm 6 sin (0-90) cos (0-90) 90-11 h 2 5 sin 90 cos 0 Sin 90 sin 0 sin (0-135) cos (6-135) 11 .5-

2 Sin 135 s1n6+cos0 Sin 135 cos 0+s|n6 sin (9-135) cos (0-135) 7 5 rspw vl l SlIl 0+ cos 6 23i) cos 0 sin 0 l .L sin (0-18Q) cos (6-180) 157.5 cos 180 sir) 0 cos 0 180-360- above functions repeat.

The bridge circuit BR40 includes a P of channels as The above table reflects not only the effects of generating described in connection with the previous figures, the specific 7 5 ta gent functions at i f f t 79 Q 9 changes necessaFy fi increments of 22.5 to accomplish bidirectional interpolations, i.e., to indicate to the system which direction to interpolate.

The same bidirectional interpolation technique can also be practiced in the other illustrated embodiments.

' Polarity reversed to meet direction control requirements.

The output of OP42 is fed to the summing amplifier OP60 via the summing resistor R49, while the output of OPSZ is coupled via summing amplifier OP53 and the interpolating divider, OP40 to the same summer OP60.

A coupling between the channels is provide, being similar to that shown in FIG. 3 and embodied as resistance R58 which couples the output of OP42 to the input of OP53. Resistances R58, R59 and R60 correspond to the resistances designated R, aR and aR in FIG. 3.

The output of OP53 is coupled to the linear, compensated interpolator IP40 which comprises a set of switches 61 through 66 with associated weighted summing resistances- R6l. These switching elements are controlled by the least significant digits of the counter CN40 to insert different combinations of resistance R61 in the input circuit of OP60 to thereby provide the fine angle a, adjustment of the interpola- As previously noted, the output of the interpolator is coupled to the summing junction of OP60 where it is combined with the signal from OP42.

The output of OP60 is fed to the null circuit NC40 and, specifically, to the phase detector 70 thereof. The detector is also supplied with a reference input via a reference amplifier 7] in conventional manner and supplies a DC of one or the other polarity above and below the null condition. W v 7 The output of the phase detector is coupled to a driver circuit 72 for supplying a voltage-controlled oscillator 74 with a driving signal when above or below the null. The oscillator 74 initiates counting operations in the counter CN40 at a rate approximately proportionedto the magnitude of the bridge error.

The output of the detector also supplies a polarity-sensitive direction control circuit 73 which, via driver 80 in the counter CN40, controls the direction of count.

The counter CN40 illustratively includes 1 1 stages of which the first seven, C 1 through C7, provide the fine angle a; control via decoder L41. The most significant digits C8 through C11 on the other hand supply the coarse processor angle adjustment via the decoder circuit L40. L40 translates the binary coded signals from stages C8 through C11 into switching signals for controlling the coarse angle function generator F640 in the bridge BR40 while L41 provides the analogous function for fine angle adjustment. Appropriate logic circuits coordinate L40 and L41.

It should be noted that the system of FIG. 4 has the ability to process DC inputs. Furthermore, the choice of operational amplifiers and feedback and logic elements in the circuit of FIG. 4 permit this system to be fabricated substantially in LS] configuration Moreover, in spite of its simplicity, the circuit produces a resolver-to-digital conversion having very low interpolation errors, there being a reduction of the error to l/64th the value which occurs in the case of an unmodified tangent technique employing a unidirectional 45 interpolation scheme. Those interpolation errors which do exist in FIG.

4 are compensated for most points by adjustment of the values of resistance R61 in the manner previously described.

Besides being applicable to the diverse functions described in the introduction, the invention, in analogy to the CT, can also serve as a data generator. For example, when has a value of 0, the input signals become simply 0 and R. The latter may be regarded as a single constant input for purposes of data generation. Under these conditions, the illustrated embodiments supply output signals representative of the processor angle only.

What is claimed is: 1. Apparatus for processing signal inputs representing angular data 0 comprising:

1. means for generating representations of coarse processor angle a 2. means generating tangent-type functions and connected to be controlled by said coarse angle generating means for generating tangent-type functions of said coarse angle,

3. a first signal computing circuit interconnecting said inputs and function generating means for processing said inputs and generated functions to generate a first signal,

4. a second signal computing circuit interconnecting said inputs and function generating means for processing said inputs and generated functions to generate a second signal,

5. means for generating representations of fine processor angle 11,,

6. interpolating means connected to said first signal computing circuit and said fine angle generating means for interpolating said first signal as a function of fine angle a; to develop a third signal, and

7. means combining said second and third signals to derive an output indicative of the relationship between 0, ct and 01;.

2. Apparatus as defined in claim 1 in which said interpolating means include a linear interpolator, said first signal is nonlinear, and wherein said first signal bears a relationship to said second signal such that interpolating errors are independent of the adjusted setting of said coarse angle generating means.

3. Apparatus as defined in claim 2 in which said relationship is such that a, varies as a function of tan(0a over at least a portion of the interpolating interval.

4. Apparatus as defined in claim 1 including switching means in said first and second signal computing circuits connected to be controlled by said coarse angle generating means to vary the interconnection of said inputs and function generating means in dependence on the adjusted value of said coarse angle.

5. Apparatus as defined in claim 1 in which said first and second computing circuits include means for producing bidirectional interpolation.

6. Apparatus as defined in claim 1 including means responsive to said combining means for signalling the sum of the adjusted values of said coarse and fine angle generating means.

7. Apparatus as defined in claim 1 in which said first and second signals each include the factor l/cosineot for one range of a and l/sina, for another range of 01,.

8. Apparatus as defined in claim 1 in which said first and second signals each include representations of a trigonometric function of (0 0 9. Apparatus as defined in claim 1 in which the increments of said coarse angle are greater than the operating range of said interpolating means.

10. Apparatus for computing the relationship between angular data 0 represented by two input signals, and processor angle data 0: represented as coarse and fine component angles, a and 04,, respectively comprising:

1. tangent function generating means including tapped divider means supplying parameters representing the tangent and cotangent values of a 2. computing means for processing said input signals by said tangent function generating means to generate two signals the ratio of which is a function of (0a but independent of 01,,

3. interpolating means controlled by a representation of a,

for interpolating one of said computed signals, and

4. means jointly responsive to said other computed signal and said interpolate signal for deriving an output indicative of the relationship between a and 6.

11. Apparatus as defined in claim 10 in which both of said computed signals are dependent in one operating range on l/cosinea and in another range on llsina but their ratio in any range is independent of a 12. Apparatus as defined in claim 10 in which said function generating means include tap selection means operable to select tangent functions at taps of said divider means through certain octants and cotangent functions at said same taps in other octants.

13. Apparatus as defined in claim 10 including means e b 19. .5? ntsrr ls Qt &9)?! the ssmh qs 9 s id tangent function generating means and said input signals.

14. Apparatus as defined in claim 10 in which one of said computed signals is a function of the sine of (a,).

15. Apparatus as defined in claim in which said function generating means include signal tap selection means controlled by said coarse angle 01,.

16. Apparatus for computing relationships between angular data 0 represented by two input signals, and processor angle data :1 represented by coarse and fine components a, and (1,, respectively. comprising:

1. tangent function generating means,

2. constant means interconnecting said function generating means and said inputs to derive first and second signals representing factors [sin(D-Aa) sin Aa] and sin Aa, respectively, where D is a constant and Aa=0a,,

3. means for interpolating said first signal, and

4. means combining said interpolated signal and said second signal for computing said relationship.

17. Apparatus as defined in claim 16 in which said first and second signals both also represent alternatively the factors l/cosa,,,,- llsina 18. Apparatus as defined in claim 16 in which said first and second signals represent the factors [sin(DAa) sinAa]/f(a and sin Aa/f(a,), respectively, where f(a is a trigonometric function of 01,.

19. Apparatus as defined in claim 16 in which said combining means comprises mean for summing said first and second signals such that when a, and a, are adjusted to null said first and second signals, then a, is proportional to 1 sin (D-Aa) sin A0;

when the effect of interpolation error correction is excluded.

20. Apparatusaccording to claim 16 including means for modifying said first signal by said second signal. g

21. Apparatus for computing relationships between angular date 0 represented by signalsfi (0) and f: (0), and processor angle data a represented by coarse and fine processor angles a, and 01,, respectively, comprising:

i summing means for deriving first and second signals,

la. means for generating functions of tana, and cotana 2. switching means controlled by the functions tana and cotana, at 45 intervals of afor selectively coupling combinations of said signals f, (9), f (0) to said summing means,

3. interpolating means connected to interpolate said first derived signal as a function of 01,, and

4. means combining said interpolated signal and said second derived signal for computing said relationships.

22. Apparatus as defined in claim 21 in which said interpolating means include a linear interpolator and in which the relationship between said first and second derived signals yields an interpolation error characteristic which is indepen' dent of the adjusted value of a...

23. Apparatus as defined in claim 22 including means for modifying said first derived signal by said second derived signal for improving said error characteristic.

24. Apparatus as defined in claim 22 including means coupled to said linear interpolator for compensating said error characteristic.

25. Apparatus as defined in claim 21 in which the interpolation interval is different from the coarse angle interval and interpolation is bidirectional. 1

26. Apparatus as defined in claim 21 in which said first and second derived signals represent factors including sin Aa where Aoz=6a,..

27. Apparatus as defined in claim 21 in which said first and second derived signals represent factors including and sinAa/fliz respectively, D Being a constant and f(a,)

polating means comprise a linear interpolator and wherein for the adjusted values of a and a the value of a, is proportional to sm (p A11) +1 as g. MW

where D is the interpolating interval and Aa=0a,.

29. Apparatus as defined in claim 21 including means for automatically adjusting said switching means and interpolating means in accordance with the output of said combining means.

30. Apparatus as defined in claim 29 in which said automatic adjustment means include a variable frequency oscillator, the frequency of which is controlled by the output of said combining means.

31. The process of deriving data from input signals representing angle 0 without the need for computation elements for generating functions having values other than zero or unity (:tl comprising the steps of:

l. generating two signals representing summations of said input signals in accordance with the zero and unity values (1-1 of tangent and cotangent of a coarse processor angle la. modifying one of said signals by the other to permit linear interpolation with reduced error,

2. linearly interpolating said modified signal as a function of a fine processor angle 01,, and

3. combining said interpolated signal and said other signal to derive said data.

32. The process as defined in claim 31 in which said two signals are generated such that the interpolation error is independent of the adjusted value of said coarse processor angle.

33. The process as defined in claim 31 in which said signal generation consists essentially of summation and inversion. 0 g

34. The process as defined in claim 31 in which one of said generated signals is modified to control the interpolation interval and in which said interpolation interval is less than the interval between said coarse processor angle values.

35. The process as defined in claim 31 in which said generated signals are combined to establish a null condition.

36. The process as defined in claim 31 in which said interpolated signal is compensated for interpolation errors.

37. The process as defined in claim 31 in which said two signals are generated such that each includes the factor l/cosa for certain values of a, and l/sina for other values.

38. The process as defined in claim 31 in which said two signals are generated such that each includes the factor sineAa where Ad=0'-1 39. The process as defined in claim 21 in which said two signals are generated and interpolated such that, when equal where Aa=0-a and D is the interpolation interval.

40. Bridge apparatus for processing angular data represented by sine and cosine signals RSin0 and RCos0, to derive related processor angle data 0:, represented by coarse and fine processor angles 01 and 01,, comprising:

1. coarse angle generating means for generating representations of a 2. fine angle generating means for generating representations of a 3. tangent function generating means adjusted by said coarse angle generating means for computing tangent a,- and cotangent a 4. switching means controlled by said coarse angle generating means for alternatively coupling said cosine and sine signals to said tangent function generating means and for deriving first and second signals therefrom,

5. first combining means for combining said first signal with said sine and cosine signals alternatively, to derive a first computation signal,

6. second combining means for combining said second signal with said sine and cosine signals alternatively and 41 Apparatus as defined in claim 40 in which said tangent function generating means comprise a tapped divider having tap selection means for supplying said first and second signals.

42. Apparatus as defined in claim 40 in which said first and second combining means each comprise transformer means.

43. Apparatus as defined in claim 40 in which said second combining means and modifying means comprise the primary and tapped secondary of transformer means, respectively, the latter also including tap selection means controlled by said coarse angle generating means.

44. Apparatus as defined in claim 40 in which said first and second signals, for certain values of a are proportional respectively to RCosUTanoz and RCosUT an (a,.+D) where D is the interval between coarse angle values.

45. Apparatus as defined in claim 40 in which said first and second computation signals respectively include representations of the factors RSin(Aa) and R[Sin (D-aA) +SinAa], where Aa=a,.

46. Apparatus as defined in claim 40 including means coupled to said interpolating means for compensating for interpolation error.

47. Bridge apparatus for processing angular data r resse x si s a 9 11 si n ls 88in! and qsiaew derive related processor angle data 1! represented by coarse and fine processor angles a and 01,, respectively, comprising:

1. coarse angle generating means for generating representations of a 2. fine angle generating means for generating representations of 01,,

3. first tangent function generating means controlled by said coarse angle generating means and connected to respond to said cosine signal to derive a first signal,

4. second tangent function generating means controlled by said coarse angle generating means and connected to respond to said sine signal to derive a second signal,

5. first combining means for combining said first signal with said sine signal to derive a first computation signal,

6. second combining means for combining said second signal with said cosine signal to derive a second computation signal,

7.v interpolating means coupled to said fine angle generating means and including a linear interpolator connected to interpolate one of said computation signals to drive an interpolated signal, and

8. detecting means for combining the other computation signal and said interpolated signal to detect the relationship between 0 and a.

48. Bridge apparatus as defined in claim 4 in which said first and second tangent function generating means each comprise tapped divider means.

49. Bridge apparatus as defined in claim 47 in which said first and second tangent function generating means each comprise a transformer having a primary energized by the respec tive sine or cosine signal and a tapped secondary having tap selection means controlled by said coarse angle generating means.

50. Apparatus as defined in claim 47 including polarity reversing switching means in each of said first and second combining means, said reversing means being controlled by said coarse angle generating means.

, 51. Apparatus as defined in claim 47 in which said first and second signals represent the factors Cosfl'l'ana, and SinUlana,

, respectively, for certain values of a 52. Apparatus as defined in claim 47 in which said first and second computation signals represent the factors SinAa/Cosa, 5 and CosAoz/Cosoz for certain values of (1,.

53. Apparatus as defined in claim 47 including means for including in the computational signal which is interpolated, a portion of the other computational signal for improving the accuracy of interpolation.

54. Apparatus as defined in claim 53 in which said modifying means supplies said portion according to the factor (1-Cos D)/Sin D, where D is a constant relating to the interpolation interval.

55. Apparatus as defined in claim 53 in which said computational signal which is interpolated includes the factor R sin (D Aa) sin Aa cos ac sin D where D represents the interpolation interval and Aa=0a 56. Apparatus as defined in claim 55 in which said other computational signal includes the factor RSinAa/Cosa,..

57. Apparatus supplied with reference input R for processing angular data comprising:

1. means for generating representations of coarse processor angle a 2. means generating tangent-type functions and connected to be controlled by said coarse angle generating means for generating tangent-type functions of said coarse angl 3. a first signal computing circuit interconnecting said input and function generating means for computing a first signal,

4. a second signal computing circuit interconnecting said input and function generating means for computing a second signal, 5. means for generating representations of fine processor angle a,,

6. interpolating means connected to said first signal computing circuit and said fine angle generating means for interpolating said first signal as a function of fine angle a, to develop a third signal, and

7. means combining said second and third signals to derive an output indicative of the relationship between R, a and (1;.

58. Apparatus as defined in claim 57 in which said interpolating means include a linear interpolator, said first signal is nonlinear, and wherein said first signal bears a relationship to said second signal such that interpolating errors are independent of the adjusted setting of said coarse angle generating means.

59. Apparatus as defined in claim 58 in which said relationship is such that a; varies as a function of tana over at least a portion of the interpolating interval.

60. Apparatus as defined in claim 57 including switching means in said first and second signal computing circuits connected to be controlled by said coarse angle generating means to vary the interconnection of said input and function generating means in dependence on the adjusted value of said coarse angle.

61. Apparatus as defined in claim 57 in which said first and second computing circuits include means for producing bidirectional interpolation.

62. Apparatus as defined in claim 57 in which said first and second signals each include the factor l/cosineoz for one range of a and l/sina for another range of 01,.

63. Apparatus as defined in claims 57 in which said first and second signals each include representations of a trigonometric function of a 64. Apparatus as defined in claim 57 in which the increments of said coarse angle are greater than the operating range of said interpolating means.

zgggg UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTIN Patent No- 47556??? DatedFebruary 29, 1972 Inventofls) Hargr P. Bifulco, Jr.

It: is certified that error appears in the above-identified patent and that said Letters Patent are hereby corrected as shown below:

Abstract, penultimate line, of, firstoccurrence, should read. or e Column 1, line 35, after "equals" insert 9.

Column 7, line 33, third ocourrence, should read Column 8, lines 35 to 37, the equations-should not run together, but should appear in column form as follows:

Column 9, line 56, "sin 90" should read sin 90 Column 12, line 68, "l/cosine should read l/cosineo( Colunm 13, line 12, delete the word "constant";

" should read l/cos 2";

Column 13, line 21, l/cos c q Column 13, line 24, "f( should read fflx Column 13, line 44, "0Q" should read et 14, line 47, l/cos shouldread l/cos and Column 15, line 60, "4" should read 47 J Signed and sealed this 18th day of July 1972.

(SEAL) Attest:

EDWARD M.FLETGHER,JR. ROBERT GOTTSCHALK Attesting Officer Comissioner of Patents (5/69) v STAS PATENT FQE QE'HFEGATE @F Patent No q filrfififi DetedFebruery 29 1972 Inventofls) Harry P. Bifulco, Jr.

It is certified that error appears in the above-identified patent and that said Letters Patent are hereby corrected as shown below:

Abstract, penultimate line, "of", first occurrence, should read. or

Column 1, line 35, after equals insert a (301 7, line 33, third occurrence, should read Column 8, lines 35 to 37, the equations should not run together, but should appear in 121% fo as follows:

Column 9, line 56 "sin 90 should read m sin 96 Col 12, line 68, llcosine should read l/cosineq Colunm 13, line 12, delete the word consrant;

Colman 13, line 21 licos should read M llcos Coiumn 13, line 24, "f( should read M f( (3011mm 1?, line 44, 1? should read i Col 14, line 47, l/cos should read 1/c0s and fliolunm 15, line 69, "4" should read 47 J Signed and sealed this 18th day of July 1972 Arrest:

EHAED MQFLETGHER JR ROBERT GOTTSCHALK Atreeting Officer Commissioner of Patents 

1. Apparatus for processing signal inputs representing angular data theta comprising:
 1. means for generating representations of coarse processor angle Alpha c,
 2. means generating tangent-type functions and connected to be controlled by said coarse angle generating means for generating tangent-type functions of said coarse angle,
 3. a first signal computing circuit interconnecting said inputs and function generating means for processing said inputs and generated functions to generate a first signal,
 4. a second signal computing circuit interconnecting said inputs and function generating means for processing said inputs and generated functions to generate a second signal,
 5. means for generating representations of fine processor angle Alpha f,
 6. interpolating means connected to said first signal computing circuit and said fine angle generating means for interpolating said first signal as a function of fine angle Alpha f to develop a third signal, and
 7. means combining said second and third signals to derive an output indicative of the relationship between theta , Alpha c and Alpha f.
 2. switching means controlled by the functions tan Alpha c and cotan Alpha c at 45* intervals of Alpha for selectively coupling combinations of said signals f1 ( theta ), f2 ( theta ) to said summing means,
 2. fine angle generating means for generating representations of Alpha f,
 2. means generating tangent-type functions and connected to be controlled by said coarse angle generating means for generating tangent-type functions of said coarse angle,
 2. constant means interconnecting said function generating means and said inputs to derive first and second signals representing factors (sin(D- Delta Alpha ) + sin Delta Alpha ) and sin Delta Alpha , respectively, where D is a constant and Delta Alpha theta - Alpha c,
 2. Apparatus as defined in claim 1 in which said interpolating means include a linear interpolator, said first signal is nonlinear, and wherein said first signal bears a relationship to said second signal such that interpolating errors are independent of the adjusted setting of said coarse angle generating means.
 2. means generating tangent-type functions and connected to be controlled by said coarse angle generating means for generating tangent-type functions of said coarse angle,
 2. computing means for processing said input signals by said tangent function generating means to generate two signals the ratio of which is a function of ( theta - Alpha c) but independent of Alpha c,
 2. linearly interpolating said modified signal as a function of a fine processor angle Alpha f, and
 2. fine angle generating means for generating representations of Alpha f,
 3. first tangent function generating means controlled by said coarse angle generating means and connected to respond to said cosine signal to derive a first signal,
 3. tangent function generating means adjusted by said coarse angle generating means for computing tangent Alpha c and cotangent Alpha c,
 3. interpolating means connected to interpolate said first derived signal as a function of Alpha f, and
 3. a first signal computing circuit interconnecting said inputs and function generating means for processing said inputs and generated functions to generate a first signal,
 3. means for interpolating said first signal, and
 3. Apparatus as defined in claim 2 in which said relationship is such that Alpha f varies as a function of tan( theta - Alpha c) over at least a portion of the interpolating interval.
 3. combining said interpolated signal and said other signal to derive said data.
 3. interpolating means controlled by a representation of Alpha f for interpolating one of said computed signals, and
 3. a first signal computing circuit interconnecting said input and function generating means for computing a first signal,
 4. switching means controlled by said coarse angle generating means for alternatively coupling said cosine and sine signals to said tangent function generating means and for deriving first and second signals therefrom,
 4. means jointly responsive to said other computed signal and said interpolate signal for deriving an output indicative of the relationship between Alpha and theta .
 4. Apparatus as defined in claim 1 including switching means in said first and second signal computing circuits connected to be controlled by said coarse angle generating means to vary the interconnection of said inputs and function generating means in dependence on the adjusted value of said coarse angle.
 4. means combining said interpolated signal and said second signal for computing said relationship.
 4. a second signal computing circuit interconnecting said inputs and function generating means for processing said inputs and generated functions to generate a second signal,
 4. means combining said interpolated signal and said second derived signal for computing said relationships.
 4. a second signal computing circuit interconnecting said input and function generating means for computing a second signal,
 4. second tangent function generating means controlled by said coarse angle generating means and connected to respond to said sine signal to derive a second signal,
 5. first combining means for combining said first signal with said sine signal to derive a first computation signal,
 5. means for generating representations of fine processor angle Alpha f,
 5. Apparatus as defined in claim 1 in which said first and second computing circuits include means for producing bidirectional interpolation.
 5. first combining means for combining said first signal with said sine and cosine signals alternatively, to derive a first computation signal,
 5. means for generating representations of fine processor angle Alpha f,
 6. interpolating means connected to said first signal computing circuit and said fine angle generating means for interpolating said first signal as a function of fine angle Alpha f to develop a third signal, and
 6. second combining means for combining said second signal with said sine and cosine signals alternatively and with a signal derived from said first combining means,
 6. second combining means for combining said second signal with said cosine signal to derive a second computation signal,
 6. Apparatus as defined in claim 1 including means responsive to said combining means for signalling the sum of the adjusted values of said coarse and fine angle generating means.
 6. interpolating means connected to said first signal computing circuit and said fine angle generating means for interpolating said first signal as a function of fine angle Alpha f to develop a third signal, and
 7. means combining said second and third signals to derive an output indicative of the relationship between theta , Alpha c and Alpha f.
 7. Apparatus as defined in claim 1 in which said first and second signals each include the factor 1/cosine Alpha c for one range of Alpha c, and 1/sin Alpha c for another range of Alpha c.
 7. interpolating means coupled to said fine angle generating means and including a linear interpolator connected to interpolate one of said computation signals to drive an interpolated signal, and
 7. means combining said second and third signals to derive an output indicative of the relationship between R, Alpha c and Alpha f.
 7. means for modifying the signal derived in said second combining means as a function of cosine Alpha c to derive a second computation signal,
 8. Apparatus as defined in claim 1 in which said first and second signals each include representations of a trigonometric function of ( theta - Alpha c).
 8. detecting means for combining the other computation signal and said interpolated signal to detect the relationship between theta and Alpha .
 8. means including a linear interpolator controlled by said fine angle generating means for interpolating said second computation signal to derive an interpolated signal, and
 9. Apparatus as defined in claim 1 in which the increments of said coarse angle are greater than the operating range of said interpolating means.
 9. detection means responsive to said first and interpolated signals to detect the relationship between theta and Alpha .
 10. Apparatus for computing the relationship between angular data theta represented by two input signals, and processor angle data Alpha represented as coarse and fine component angles, Alpha c and Alpha f, respectively comprising:
 11. Apparatus as defined in claim 10 in which both of said computed signals are dependent in one operating range on 1/cosine Alpha c and in another range on 1/sin Alpha c but their ratio in any range is independent of Alpha c.
 12. Apparatus as defined in claim 10 in which said function generating means include tap selection means operable to select tangent functions at taps of said divider means through certain octants and cotangent functions at said same taps in other octants.
 13. Apparatus as defined in claim 10 including means operable at 45* intervals of Alpha c to vary the combining of said tangent function generating means and said input signals.
 14. Apparatus as defined in claim 10 in which one of said computed signals is a function of the sine of ( theta - Alpha c).
 15. Apparatus as defined in claim 10 in which said function generating means include signal tap selection means controlled by said coarse angle Alpha c.
 16. Apparatus for computing relationships between angular data theta represented by two input signals, and processor angle data Alpha represented by coarse and fine components Alpha c and Alpha f, respectively, comprising:
 17. Apparatus as defined in claim 16 in which said first and second signals both also represent alternatively the factors 1/cos Alpha c, and 1/sin Alpha c.
 18. Apparatus as defined in claim 16 in which said first and second signals represent the factors (sin(D- Delta Alpha ) + sin Delta Alpha )/f( Alpha c) and sin Delta Alpha /f( Alpha c), respectively, where f( Alpha 2) is a trigonometric function of Alpha c.
 19. Apparatus as defined in claim 16 in which said combining means comprises mean for summing said first and second signals such that when Alpha c and Alpha f are adjusted to null said first and second signals, then Alpha f is proportional to when the effect of interpolation error correction is excluded.
 20. Apparatus according to claim 16 including means for modifying said first signal by said second signal.
 21. Apparatus for computing relationships between angular date theta represented by signals f1 ( theta ) anD f2 ( theta ), and processor angle data Alpha represented by coarse and fine processor angles Alpha c and Alpha f, respectively, comprising:
 22. Apparatus as defined in claim 21 in which said interpolating means include a linear interpolator and in which the relationship between said first and second derived signals yields an interpolation error characteristic which is independent of the adjusted value of Alpha c.
 23. Apparatus as defined in claim 22 including means for modifying said first derived signal by said second derived signal for improving said error characteristic.
 24. Apparatus as defined in claim 22 including means coupled to said linear interpolator for compensating said error characteristic.
 25. Apparatus as defined in claim 21 in which the interpolation interval is different from the coarse angle interval and interpolation is bidirectional.
 26. Apparatus as defined in claim 21 in which said first and second derived signals represent factors including sin Delta Alpha where Delta Alpha theta - Alpha c.
 27. Apparatus as defined in claim 21 in which said first and second derived signals represent factors including and sin Delta Alpha /f( Alpha c), respectively, D being a constant and f( Alpha c) being alternatively cos Alpha c and sin Alpha c.
 28. Apparatus as defined in claim 21 in which said interpolating means comprise a linear interpolator and wherein for the adjusted values of Alpha c and Alpha f, the value of Alpha f is proportional to where D is the interpolating interval and Delta Alpha theta - Alpha c.
 29. Apparatus as defined in claim 21 including means for automatically adjusting said switching means and interpolating means in accordance with the output of said combining means.
 30. Apparatus as defined in claim 29 in which said automatic adjustment means include a variable frequency oscillator, the frequency of which is controlled by the output of said combining means.
 31. The process of deriving data from input signals representing angle theta without the need for computation elements for generating functions having values other than zero or unity ( + or - 1) comprising the steps of:
 32. The process as defined in claim 31 in which said two signals are generated such that the interpolation error is independent of the adjusted value of said coarse processor angle.
 33. The process as defined in claim 31 in which said signal generation consists essentially of summation and inversion.
 34. The process as defined in claim 31 in which one of said generated signals is modified to control the interpolation interval and in which said interpolation interval is less than the interval between said coarse processor angle values.
 35. The process As defined in claim 31 in which said generated signals are combined to establish a null condition.
 36. The process as defined in claim 31 in which said interpolated signal is compensated for interpolation errors.
 37. The process as defined in claim 31 in which said two signals are generated such that each includes the factor 1/cos Alpha c for certain values of Alpha c and 1/sin Alpha c for other values.
 38. The process as defined in claim 31 in which said two signals are generated such that each includes the factor sine Delta Alpha where Delta Alpha theta - Alpha c.
 39. The process as defined in claim 21 in which said two signals are generated and interpolated such that, when equal where Delta Alpha theta - Alpha c and D is the interpolation interval.
 40. Bridge apparatus for processing angular data represented by sine and cosine signals RSin theta and RCos theta , to derive related processor angle data Alpha , represented by coarse and fine processor angles Alpha c and Alpha f, comprising:
 41. Apparatus as defined in claim 40 in which said tangent function generating means comprise a tapped divider having tap selection means for supplying said first and second signals.
 42. Apparatus as defined in claim 40 in which said first and second combining means each comprise transformer means.
 43. Apparatus as defined in claim 40 in which said second combining means and modifying means comprise the primary and tapped secondary of transformer means, respectively, the latter also including tap selection means controlled by said coarse angle generating means.
 44. Apparatus as defined in claim 40 in which said first and second signals, for certain values of Alpha c, are proportional respectively to RCos theta Tan Alpha c and RCos theta Tan ( Alpha c+D) where D is the interval between coarse angle values.
 45. Apparatus as defined in claim 40 in which said first and second computation signals respectively include representations of the factors RSin( Delta Alpha ) and R(Sin (D- Alpha Delta ) +Sin Delta Alpha ), where Delta Alpha theta - Alpha c.
 46. Apparatus as defined in claim 40 including means coupled to said interpolating means for compensating for interpolation error.
 47. Bridge apparatus for processing angular data represented by sine and cosine signals RSin theta and RCosine theta to derive related processor angle data Alpha represented by coarse and fine processor angles Alpha c and Alpha f, respectively, comprising:
 48. Bridge apparatus as defined in claim 4 in which said first and second tangent function generating means each comprise tapped divider means.
 49. Bridge apparatus as defined in claim 47 in which said first and second tangent function generating means each comprise a transformer having a primary energized by the respective sine or cosine signal and a tapped secondary having tap selection means controlled by said coarse angle generating means.
 50. Apparatus as defined in claim 47 including polarity reversing switching means in each of said first and second combining means, said reversing means being controlled by said coarse angle generating means.
 51. Apparatus as defined in claim 47 in which said first and second signals represent the factors Cos theta Tan Alpha c and Sin theta Tan Alpha c, respectively, for certain values of Alpha c.
 52. Apparatus as defined in claim 47 in which said first and second computation signals represent the factors Sin Delta Alpha /Cos Alpha c and Cos Delta Alpha /Cos Alpha c for certain values of Alpha c.
 53. Apparatus as defined in claim 47 including means for including in the computational signal which is interpolated, a portion of the other computational signal for improving the accuracy of interpolation.
 54. Apparatus as defined in claim 53 in which said modifying means supplies said portion according to the factor (1-Cos D)/Sin D, where D is a constant relating to the interpolation interval.
 55. Apparatus as defined in claim 53 in which said computational signal which is interpolated includes the factor where D represents the interpolation interval and Delta Alpha theta - Alpha c.
 56. Apparatus as defined in claim 55 in which said other computational signal includes the factor RSin Delta Alpha /Cos Alpha c.
 57. Apparatus supplied with reference input R for processing angular data comprising:
 58. Apparatus as defined in claim 57 in which said interpolating means include a linear interpolator, said first signal is nonlinear, and wherein said first signal bears a relationship to said second signal such that interpolating errors are independent of the adjusted setting of said coarse angle generating means.
 59. Apparatus as defined in claim 58 in which said relationship is such that Alpha f varies as a function of tan Alpha c over at least a portion of the interpolating interval.
 60. Apparatus as defined in claim 57 including switching means in said first and second signal computing circuits connected to be controlled by said coarse angle generating means to vary the interconnection of said input and function generating means in dependence on the adjusted value of said coarse angle.
 61. Apparatus as defined in claim 57 in which said first and second computing circuits include means for producing bidirectional interpolation.
 62. Apparatus as defined in claim 57 in which said first and second signals each include the factor 1/cosine Alpha c for one range of Alpha c, and 1/sin Alpha c for another range of Alpha c.
 63. Apparatus as defined in claims 57 in which said first and second signals each include representations of a trigonometric function of Alpha c.
 64. Apparatus as defined in claim 57 in which the increments of said coarse angle are greater than the operating range of said interpolating means. 